A family Of Cyclic Codes Over Finite Chain Rings
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1 The Islamic University of Gaza Deanery of Higher Studies Faculty of Science Department of Mathematics A family Of Cyclic Codes Over Finite Chain Rings Presented by: Sanaa Yusuf Sabouh Supervised by: Dr.: Mohammed Mahmoud AL-Ashker SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE AT ISLAMIC UNIVERSITY GAZA, PALESTINE 2008
2 DEDICATION To My Parents My sincere friends and to all knowledge seekers i
3 Contents DEDICATION Table of Contents Acknowledgements Abstract i ii iv v Introduction 1 1 Preliminaries Algebraic preliminaries Basic definitions of coding theory Cyclic codes over finite fields Encoding and decoding of cyclic codes About Galois rings Cyclic codes over rings of four elements Background Cyclic codes over Z Self-dual codes over Z Cyclic codes over R 2,2 = F 2 + uf Self-dual codes over R 2,2 = F 2 + uf Cyclic codes over rings of higher orders Introduction Cyclic codes over Z p m Cyclic codes over R k,p = F p + uf p u k 1 F p Idempotents of cyclic codes over Rings of four elements Generating idempotents of cyclic codes over Z Generating idempotents of cyclic codes over R 2,2 = F 2 + uf ii
4 4.3 Examples Idempotents of cyclic codes over rings of higher orders Generating idempotents of cyclic codes over Z p m Example on Generating idempotents of cyclic codes over Z 8 and Z Generating idempotents of cyclic codes over R k,p = F p + uf p u k 1 F p Example on Generating idempotents of cyclic codes over F 2 + uf 2 + u 2 F 2 and F 3 + uf Conclusion 94 Bibliography iii
5 Acknowledgements First of all, gratitude and thanks to Almighty Allah who always helps and guides me. I wish to extend my gratitude and appreciation to my supervisor Dr. Mohammed M. AL-Ashker for his help and advice during the preparation of the thesis. Thanks are also due to the examiners committee Dr. Fayik EL-Naoqe, and Dr. Arwa Ashour. Thanks are also due to the head and the staff members of the Mathematics Department, and the Faculty of Science of the Islamic University. iv
6 Abstract Codes over finite rings have received much attention recently after it was proved that important families of binary non-linear codes are images under a Gray map of linear codes over Z 4. A set of n-tuples over a ring R is called a code over R if it is an R-module. A cyclic codes of length n over the ring R is a linear code with property that if the codeword (c 0, c 1,..., c n 1 ) C then the cyclic shift (c 1, c 2,..., c 0 ) C. The cyclic codes are ideals in the ring R n = R[x]/(x n 1). A commutative ring R with identity 1 0 is called a finite chain ring if its ideals are linearly ordered by inclusion. We study in this thesis to study cyclic codes over finite chain rings. We first give a survey study a bout cyclic codes over the rings Z p k of integers modulo p k for a prime p and k 1, in particular Z 4 and Z 8 and study their structures. We will extend this study to cyclic codes for more chain rings F p + uf p, F p + uf p + u 2 F p and F p + uf p u k 1 F p for different prime number p and we will define and construct idempotent generators for cyclic over these rings and study their properties. v
7 Introduction The beginning of coding theory goes back to the middle of the last century with the work of Shannons, Hamming, Golay and others. Historically coding theory originated as the mathematical foundation for the transmission of messages over noisy channels. In fact a multitude of diverse applications have been discovered such as the minimization of noise from compact disc recordings the transmission of financial information a cross telephone lines, data transfer from one computer to another and so on. Coding theory deals with the problem of detecting and correcting transmission errors caused by noise on the channel. Mathematical background was at the beginning very little but with passing of time, various mathematical tools, such as group theory, ring theory, and linear programming have been applied to coding theory. Thus, coding theory has now become an active part of mathematical research. In many cases, the information to be sent is transmitted by a sequence of zeros and ones called binary codes, which means that the code is defined on the field {0, 1}. In this thesis we study that any Z 4 -cyclic code C has generators of the form (fh, 2fg) where fgh = x n 1 over Z 4, and C = 4 degg 2 degh. We also study that C has generators of the form (g h, 2f g ) and show that a Z 4 -cyclic code has an idempotent generator. The structure of cyclic codes over Z p m was obtained by Galderbank and sloane in [4], and later on, with a different proof by kanwar in [18]. Using the techniques presented in [18], Wan [26] extended Kanwars results to cyclic codes over Galois rings. Cyclic self dual codes and linear simplex codes over F 2 + uf 2 have been extensively studied in the papers [2], [3]. Codes over F p + uf p u k 1 F p have discussed by of a number of authors in [16]. The material of this thesis lies in five chapters. Chapter 1:- Includes algebraic preliminaries, Basic definitions of coding theory, cyclic codes over finite fields, about Galois rings that are needed in 1
8 the thesis. Chapter 2:- Include generating and dual of cyclic codes over finite chain rings of four elements as Z 4 and F 2 + uf 2, where u 2 = 0 Chapter 3:- Presents a brief introduction to codes over Z p m and F p + uf p u k 1 F p, cyclic codes over Z p m and F p + uf p u k 1 F p, where u k = 0 and examples. Chapter 4:- In this chapter we will study idempotent generators of cyclic codes over rings of four elements and present some examples. Chapter 5:- Includes generating idempotents of cyclic codes over Z p m, examples on generating idempotents of cyclic codes over Z 4 and Z 8, generating idempotents of cyclic codes over F p + uf p u k 1 F p, where u k = 0 and also we give some examples on generating idempotents of cyclic codes over F 2 + uf 2 + u 2 F 2 and F 3 + uf 3, where u 2 = 0. 2
9 Chapter 1 Preliminaries 1.1 Algebraic preliminaries The purpose of this section is to review some basic facts that will be needed through the thesis. Rings and Fields Definition [23] A nonempty set R, together with two binary operations + and. is said to form a Ring, for all a, b, c R if the following axioms are satisfied : (i) a + (b + c) = (a + b) + c (ii) a + b = b + a (iii) some element 0 (called zero) in R s.t., a + 0 = 0 + a = a (iv) for each a R, an element ( a) R, s.t., a + ( a) = ( a) + a = 0 (v) a.(b.c) = (a.b).c (vi) a.(b + c) = a.b + a.c (b + c).a = b.a + c.a Definition [23] A ring R is called a commutative ring if ab = ba for all a, b R. Again if a unique element e R s.t., ae = ea = a for all a R we say, R is a ring with unity. Unity is generally denoted by 1 (it is also called unit element or multiplicative identity). 3
10 Definition [23] An element a in a ring R with unity, is called invertible (or a unit) with respect to multiplication if some b R such that ab = 1 = ba. Definition [23] Let R be a ring. An element 0 a R is called a zero-divisor, if an element 0 b R s.t., ab = 0 or ba = 0. Definition [23] A commutative ring with unity is called an integral domain if ab = 0 in R = either a = 0 or b = 0. In other words, a commutative ring is called an integral domain if R has no zero divisors. Definition [19] A field is a nonempty set F of elements with two binary operations + (called addition) and. (called multiplication) satisfying the following axioms. For all a, b, c F: (i) F is closed under + and..; i.e., a + b and a.b are in F. (ii) Commutative laws: a + b = b + a, a.b = b.a. (iii) Associative laws: (a + b) + c = a + (b + c), a.(b.c) = (a.b).c. (iv) Distributive law: a.(b + c) = a.b + a.c. Furthermore, two distinct identity elements 0 and 1 (called the additive and multiplicative identities, respectively) must exist satisfying the following: (v) a + 0 = a for all a F. (vi) a.1 = a and a.0 = 0 for all a F. (vii) For any a in F, there exist an additive inverse element ( a) in F such that a + ( a) = 0. (viii) For any a 0 in F, there exists a multiplicative inverse element a 1 in F such that a.a 1 = 1. We usually write a.b simply as ab, and denote by F the set F {0}. Definition [23] A ring R with unity is called a Division ring or a skew field if all non zero elements of R have multiplicative inverse. Definition [23] A commutative division ring is called a field. Lemma [23] A finite integral domain is a field. Corollary [23] Z p the set of integers mod p is a field, for a prime integer p. 4
11 Subring and the characteristic of a ring Definition [23] A non empty subset S of a ring R is said to be a subring of R if S forms a ring under the binary compositions of R. The ring < Z, +,. > of integers is a subring of the ring < R, +,. > of real number. If R is a ring then 0 and R are always subrings of R, called trivial subring of R. Theorem [23] A non empty subset S of a ring R is a sub-ring of R if and only if a, b S, then ab, a b S. Definition [23] Let R be a ring. If there exists a positive integer n such that na = 0 for all a R, then R is said to have finite characteristic and also the smallest such positive integer n is called the characteristic of R. If no such positive integer exists then R is said to have characteristic zero (or infinity). Characteristic of R is denoted by char R or chr. Example (i) The characteristics of Q, R, C are 0, where Q is the set of all rational numbers, R is the set of all real numbers and C is the set of all complex numbers. (ii) The characteristic of the field Z p is p for any prime p. Ideals Definition [8] A nonempty subset I of a ring R is called a left ideal if (i) For all a, b I both a+b and a-b belong to I. (ii) For all a I and all r R ra I. Symmetrically, we define a right ideal. A nonempty subset which is both a left and a right ideal is called an ideal, or sometimes, for the sake of emphasis, a two-sided ideal. In a commutative ring the distinction between a left and a right ideal disappears. From condition (i) above it is clear that every left (or right) ideal is a subring. However, the converse need not be true. For example, in the ring Q of rational numbers, the set Z of integers is a proper subring, but not an ideal because 1 Q, 3 Z. But 3. 1 Z. In any ring, the set {0} 2 2 consisting of the zero element alone is a two-sided ideal. It is called the zero ideal and denoted by {0}. Similarly, the whole ring R is a two-sided ideal. If possesses an identity e, then R is called a unit ideal and is denoted by (e). The two ideals {0} and R are said to be improper, any ideal other than {0} and R is said to be proper. Theorem [8] If R is a ring with unity, and I is an ideal of R containing a unit, then I = R. 5
12 Definition [8]( Normal Subgroup) A subgroup H of a group G is a normal if its left and right cosets coincide, that is, if gh = Hg for all g G denoted H G, or if and only if xhx 1 H x G. Every subgroup of an Abelian group is normal center of a group Z(G) is normal. Quotient Rings Let R be a ring and let I be an ideal in the ring R. Since a, b I = a b I, we find I is a subgroup of < R, + >. Again as < R, + > is Abelian, I is a normal subgroup of R and thus we can talk of R/I, the quotient group; R/I = {I + r : r R} = set of all cosets of I in R (clearly left or right cosets are equal).we know R/I forms a group under addition defined by (r + I) + (s + I) = (r + s) + I. We now define a binary composition (product) on R/I by (r + I)(s + I) = rs + I. It is a routine exercise to check that this product is well defined on R/I. Since (a + I)[(b + I)(c + I)] = (a + I)(bc + I) = a(bc) + I = (ab)c + I = (ab + I)(c + I) = [(a + I)(b + I)](c + I). Associativity holds with respect to this product. Again, as (a + I)[(b + I) + (c + I)] = (a + I)(b + c + I) = a(b + c) + I = (ab + ac) + I = (ab + I) + (ac + I) = (a + I)(b + I) + (a + I)(c + I) We find distributivity holds. Similarly one can check that right distributivity also holds in R/I and hence R/I forms a ring, called the quotient ring or residue class ring of R by I. More information can be found in [23]. Prime Ideals and Maximal Ideals We have seen that if R is a ring and I is an ideal in R, then the set R/I of residue classes forms a ring. A very natural question arises: When is R/I an integral domain or a field? The answer to this question leads us to the following specializations of ideals. Definition [8] An ideal I R in a commutative ring R is a prime ideal if ab I implies that either a I or b I for every a, b R Definition [23] Let R be a ring. An ideal M R of R is called a maximal ideal of R if whenever A is an ideal of R such that, M A R then either A = M or A = R. 6
13 Example [23] (i) A field F has only ideals F and {0}. We can see that {0} is the only maximal ideal of F. (ii) {0} in the ring Z of integers is a prime ideal as ab {0} ab = 0 a 0 or b 0. It is an example of a prime ideal which is not maximal because {0} 2Z Z. (iii) H 4 = {4n, n Z} we can see that it is a maximal ideal in the ring E of even integers. H 4, however, is not a prime ideal as 2.2 = 4 H 4 but 2 is not belong H 4. And also is not maximal ideal because 4Z 2Z Z. In fact, H 4 is neither a maximal nor a prime ideal in Z. In the following two theorems we give alternative criterions for an ideal in an arbitrary commutative ring to be prime or maximal. Theorem [8] Let R be a commutative ring with unity, and let I R be an ideal in R. Then R/I is an integral domain if and only if I is prime ideal in R. Theorem [23] Let R be a commutative ring with unity. An ideal M of R is maximal ideal of R if and only if R/M is a field. Corollary [8] Every maximal ideal in a commutative ring R with unity is a prime ideal, but the converse is not true see Example (ii). Definition [23] Two ideals A and B are called comaximal if A + B = R. Definition [24] An ideal I of a ring R is called a primary ideal provided ab I implies that either a I or b r I for some positive integer r. Every prime is primary, but the converse is not true, 4Z is primary but not prime. Definition [8] An ideal I of a ring R is called a principal ideal if there exists an element g I such that I =< g >, where < g >= {gr : r R}. The element g is called a generator of I and I is said to be generated by g. A ring R is called a principal ideal ring if every ideal of R is principal. 7
14 Example [8] The ideal < x > in F[x] consist of all polynomials in F[x] having zero constant term. Z is a principal ideal domain. Moreover, given any nonzero ideal I of Z, the smallest positive integer in I is a generator for the ideal. Definition [24] A local ring is a ring that has a unique maximal ideal. Definition [25] A commutative ring with unity is called a chain ring if all its ideals from a chain under inclusion. R is a commutative integral domain such that any strictly ascending chain of ideal is finite. i.e. I 0 I 1 I 2... I n... must stop after finitly many ideals. Theorem [12] For a finite commutative ring R the following conditions are equivalent : i) R is a local ring and the maximal M of R is principal; ii) R is a local principal ideal ring; iii) R is a chain ring. Homomorphisms and Isomorphisms Definition [8] Let R and R be rings (or fields). A function ψ:r R is a homomorphism if for all a, b R, ψ(a + b) = ψ(a) + ψ(b) and ψ(ab) = ψ(a)ψ(b). Definition [8] An isomorphism ψ: R R that is one-to-one and onto R. is a homomorphism Definition [8] Let f:r R be a homomorphism, we define kernel of f by where 0 is a zero of R. ker f = {x R : f(x) = 0 } Theorem [8] If f:r R is a homomorphism, then ker f is an ideal of R. ker f =< 0 > if and only if f is one-one. 8
15 Rings of Polynomials Definition [8] Polynomial Let R be a ring. A polynomial f(x) with coefficients in R is an infinite formal sum i=0 a ix i = a 0 + a 1 x a n x n +..., where a i R and a i = 0 for all but a finite number of values of i. The a i are coefficients of f(x). If for some i 0 it is true that a i 0, the largest such value of i is the degree of f(x). If all a i = 0, then the degree of f(x) is undefined. Let us agree that if f(x) = a 0 + a 1 x a n x n +... has a i = 0 for i > n, then we may denote f(x) by a 0 + a 1 x a n x n. Addition and multiplication of polynomials with coefficients in a ring R are defined in a way familiar to us. Let f(x) = a 0 + a 1 x a m x m, a i R, g(x) = b 0 + b 1 x b n x n, b i R, be two polynomials over R, then we say f(x) = g(x) if m = n and a i = b i for all i. Again, addition of polynomials f(x) and g(x) is defined by f(x) + g(x) = (a 0 + b 0 ) + (a 1 + b 1 )x + (a 2 + b 2 )x Product is also defined in the usual way f(x)g(x) = (a 0 + a 1 x a m x m )(b 0 + b 1 x b n x n ) = a 0 b 0 + (a 1 b 0 + a 0 b 1 )x +... = c 0 + c 1 x + c 2 x c m+n x m+n where c k = a 0 b k + a 1 b k a k b 0 = k r=0 a rb k r Let now R[x] be the set of all polynomials over R. Zero of the ring will be the zero polynomial O(x) = 0 + 0x + 0x Additive inverse of f(x) = a 0 +a 1 x+...+a m x m will be the polynomial f(x) = a 0 a 1 x ( a m )x m. In fact, if R has unity 1 then the polynomial e(x) = 1 + 0x + 0x will be unity of R[x]. e(x) is also sometimes denoted by 1. Instead of a ring R if we start with a field F we get the corresponding ring F[x] of polynomials, see [23]. Theorem [23] Let R[x] be the ring of polynomials over a ring R, then (i) R is commutative if and only if R[x] is commutative. (ii) R has unity if and only if R[x] has unity. 9
16 Theorem [23] Let R[x] be the ring of polynomial of a ring R and suppose f(x) = a 0 + a 1 x a m x m, g(x) = b 0 + b 1 x b n x n, are two non zero polynomials of degree m and n respectively, then (i) If R is an integral domain, deg(f(x)g(x)) = m + n. (ii) R is an integral domain if and only if R[x] is an integral domain. (iii) If F is a field, F[x] may not be field. Definition [8] Let f(x) and g(x) be polynomials over the field F. If gcd(f(x), g(x)) = 1, we say that f(x) and g(x) are relatively prime (over F). In particular, f(x) and g(x) are relatively prime if and only if there exist polynomials a(x) and b(x) over F for which a(x)f(x) + b(x)g(x) = 1. Definition [8] A polynomial f(x) R[x], is monic provided its leading coefficient is 1. Definition [24] Two polynomials f and g in R[x] are called coprime, or relatively prime if R[x] =< f > + < g >. Definition [24] A polynomial f(x) R[x] is primary if the principal ideal < f(x) >= {f(x)g(x), g(x) R[x]} is primary ideal. Theorem [8] Every nonzero prime ideal of a principal ideal domain is maximal ideal. Theorem [8] If the ring F is a field, then every ideal in F[x] is principal ideal domain. Definition [23] A nonconstant polynomial f(x) F[x] is irreducible if whenever f(x) = p(x)q(x), then one of p(x) or q(x) must be constant. Definition [8] Let p(x) is irreducible polynomial in F[x] and p(x) r(x)s(x), for r(x), s(x) F[x], then either p(x) r(x) or p(x) s(x). Theorem [8] An ideal I =< p(x) > 0 in F[x] is maximal if and only if p(x) is irreducible over F. Theorem [8] Let F be a field, then the ring E = F[x]/< p(x) > is a field if and only if p(x) is irreducible. 10
17 Definition [23] Every nonconstant polynomial in F[x] can be written uniquely (up to order) as a product of irreducible polynomials. Prime and Irreducible Elements Definition [23] Let R be a commutative ring with unity, then a, b R are called associate if b = ua for some unit u in R. Definition [23] Let R be a commutative ring with unity. An element p R is called a prime element if (i) p 1, p is not a unit. (ii) For any a, b R, if p ab then p a or p b. An element p R is called an irreducible element if (i) p 0, p is not a unit. (ii) Whenever p = ab, then one of a or b must be a unit. Theorem [23] In a P ID an element is prime if and only if it is irreducible. Example [23] In the ring < Z, +,. > of integers, every prime number is a prime element as well as irreducible element. Corollary [23] In an integral domain with unity, every prime element is irreducible. The converse is not true. Example [23] Consider the ring Z 6 = {0, 1, 2, 3, 4, 5} mod 6. 2 is a prime element in Z 6 but is not irreducible. Because 2 4 = 2, where neither 2 nor 4 is a unit, we find 2 is not irreducible. (Note, Z 6 is not an integral domain) Unique Factorization Domain Definition [23] Let R be an integral domain with unity then R is called a unique factorization domain (UF D) if (i) every nonzero, non unit element a of R can be expressed as a product of finite number of irreducible elements of R and 11
18 (ii) if a = p 1 p 2... p m a = q 1 q 2... q n where p i and q j are irreducible in R then m = n and each p i is an associate of some q j. Example [23] The ring < Z, +,. > of integers is a UF D. A field < F, +,. > is always a UF D as it contains no non zero, non unit elements. Vector spaces over finite fields Definition [23] Let F q be the finite field of order q. A nonempty set V, together with some (vector) addition denoted + and scalar multiplication by elements of F q, is a vector space (or linear space) over F q if it satisfies all of the following conditions. For all u, v, w V and for all λ, µ F q : (i) u + v V ; (ii) (u + v) + w = u + (v + w); (iii) there is an element 0 V with the property 0 + v = v + 0 for all v V ; (iv) for each u V there is an element of V, called u, such that u + ( u) = 0 = ( u) + u; (v) u + v = v + u; (vi) λv V ; (vii) λ(u + v) = λu + λv, (λ + µ)u = λu + µu; (viii) (λµ)u = λ(µu); (ix) if 1 is the multiplicative identity of F q, then 1u = u. Definition [19] A nonempty subset C of a vector space V is a subspace of V if is itself a vector space with the same vector addition and scalar multiplication as V. Modules and Submodules Definition [6] Let R be any ring, and let M be an Abelian group, then M is called a left R-module if there exists a scalar multiplication ψ : R M M denoted by ψ(r, m) = rm, for all r R and all m M, such that for all r, r 1, r 2 R and all m, m 1, m 2 M, 12
19 (i) r(m 1 + m 2 ) = rm 1 + rm 2 (ii) (r 1 + r 2 )m = r 1 m + r 2 m (iii) r 1 (r 2 m) = (r 1 r 2 )m (iv) 1m = m. To denote that M is a left R-modulo. Example [6] If R is a ring then R is an R-module (Left R-module and right R-module). Vector spaces over F are F-modules where F is a field. Definition [6] Any subset of M that is a left R-module under operations induced from M is called a submodule. The subset {0} is called the trivial submodule. The module M is a submodule of itself. i.e. If M is a left R-module, then a subset N M is a submodule if and only if it is nonempty, closed under sums, and closed under multiplication by elements of R. New Ring from old Let < R, +,. > and < S, +,. > be two rings, their product is the ring (R S, +,.) whose underlying set is the Cartesian product of R and S and whose binary operations are defined by (r 1, s 1 ) + (r 2, s 2 ) = (r 1 + r 2, s 1 + s 2 ) (r 1, s 1 ).(r 2, s 2 ) = (r 1 r 2, s 1 s 2 ) This ring is called the direct product of R and S. One can similarly extend the definition to product of more than two rings. R and S are called the component rings of the direct product. Theorem [8] Z m Z n is isomorphic to the ring Z mn if and only if gcd(m, n) = 1 Theorem [8] Let m = m 1.m 2... m r where gcd(m i, m j ) = 1 if i j. Then Z m1 Z m2... Z mr is a ring isomorphic to Z m. Theorem [23] Chinese Remainder Theorem Let R be a commutative ring with unity and let I 1 and I 2 be two ideals of R, then (i) ϕ: R R/I 1 R/I 2, such that, ϕ(x) = (x + I 1, x + I 2 ) is a homomorphism such that, Kerϕ = I 1 I2. 13
20 (ii) If ϕ is onto, then I 1 and I 2 are comaximal ideals of R. Proof. (i) We leave it for the reader to verify that ϕ is homomorphism. Since x Kerϕ ϕ(x) = (I 1, I 2 ) (x + I 1, x + I 2 ) = (I 1, I 2 ) x + I 1 = I 1 and x + I 2 = I 2 x I 1 and x I 2 x I 1 I2 we find Kerϕ = I 1 I2. (ii) Suppose ϕ is onto. Then given (1 + I 1, 0 + I 2 ) R/I 1 R/I 2, x R, s.t., ϕ(x) = (1 + I 1, I 2 ) = (x + I 1, x + I 2 ) = (1 + I 1, I 2 ) = x + I 1 = 1 + I 1 and x + I 2 = I 2 = 1 x I 1 and x I 2 = (1 x) + x I 1 + I 2 = 1 I 1 + I 2 = I 1 + I 2 = R or that I 1 and I 2 are comaximal. Finite Fields The order of a field is the number of elements in the field. If the order is infinite, we call the field an infinite field, and if the order is finite, we call the field a finite field or a Galois field. Definition [23] A finite field with p m elements is called a Galois field of order p m and is denoted by GF (p m ). Theorem [23] For any prime p and any positive integer m, there exists a finite field, unique up to isomorphism, with q = p m elements. Lemma [19] For every element β of a finite field F with q elements, we have β q = β. Definition [8] The order of a nonzero element α F q, denoted by ord(α), is the smallest positive integer k such that α k = 1. Definition [23] In an Abelian group. If a has order n and b has order m with gcd(m, n) = 1 them ab has order mn Definition [8] (primitive Root of Unity) An element α of a field is an nth root of unity if α n = 1. It is a primitive nth root of unity if α n = 1 and α m 1 for 0 < m < n. An element α in a finite field F q is called a primitive element (or a generator) of F q if F q = {0, α, α 2,..., α q 1 }. Theorem [24] The elements of F q are precisely the roots of the polynomial x q x. 14
21 Theorem [8] Division Algorithm Let f(x) and g(x) be in F q [x], where F q [x] is the ring of all polynomials over the field F q with g(x) nonzero, then 1. There exist unique polynomials h(x), r(x) F q [x], such that f(x) = g(x)h(x) + r(x), where deg r(x) < deg g(x) or r(x) = If f(x) = g(x)h(x) + r(x), then gcd(f(x), g(x)) = gcd(g(x), r(x)). Corollary [23] Let f(x) F[x], then α is root of f(x) if and only if x α is a factor of f(x) over F Theorem [24] (Hensels Lemma) Let f(x) Z 4 [x]. Suppose µ(f(x)) = h 1 (x)h 2 (x)... h k (x), where h 1 (x), h 2 (x),..., h k (x) are pairwise coprime polynomials in F 2 [x]. Then there exist g 1 (x), g 2 (x),..., g k (x) Z 4 [x] such that: 1. µ(g i (x)) = h i (x) for 1 i k, 2. g 1 (x), g 2 (x),..., g k (x) are pairwise coprime, and 3. f(x) = g 1 (x)g 2 (x)...g k (x). Extension Field Definition [8] (Extension Field) A field E is called an extension of a field F if F E. Thus R is an extension field of Q and C is an extension field of both R and Q. Theorem [8] Let F be a field, and let f(x) F[x] be a nonconstant polynomial. Then there exist an extension E of F and α E such that f(α) = 0 Example [8] Let F = R, and let f(x) = x 2 + 1, which is well known to have no zeros in R and thus is irreducible over R by Theorem Then < x > is a maximal ideal in R[x], so R[x]/ < x > is a field. Identifying r R with r+ < x > in R[x]/ < x >, we can view R as a subfield of E = R[x]/ < x >. Let α = x+ < x >. Computing in R[x]/< x >, we find < α >= (x+ < x >) 2 + (1+ < x >) =< x > + < x >= 0. Thus α is a zero of x
22 Definition [8](Algebraic, Transcendental) An element α of an extension field E of a field F is algebraic over F if f(α) = 0 for some nonzero f(x) F[x]. If α is not algebraic over F, then α is transcendental over F. extension field of Q. algebraic element over Q. C is an Since 2 is a zero of x 2 2, we see that 2 is an Also, i is an algebraic element over Q, being a zero of x It is well known (but not easy to prove) that the real numbers π and e are transcendental over Q. Here e is the base for the natural logarithms. Definition [8] If a polynomial f(x) F[x] factors into linear factors f(x) = a(x α 1 )(x α 2 )...(x α n ) over an extension field K, we say that f(x) splits over K. Definition [23] Let f(x) F[x]. A splitting field for f(x) is an extension field K of F with the property that f(x) splits over K, f(x) = β(x α 1 )(x α 2 )...(x α n ) and that K = F(α 1,..., α n ). Take for instance, f(x) = x Q[x], then as x = (x + i)(x i), we find splitting field of f(x) over Q will be Q(i). However if f(x) = x is taken as a polynomial over R, then its splitting field over R is R(i) = C the field of complex numbers. Theorem [23] Every polynomial f(x) F[x] has a splitting field, and any two splitting fields for f(x) are isomorphic. Definition [23] The degree of the extension K of the field F, denoted by [K : F], is the dimension of K as a vector space over F. K is called a finite extension if [K : F] is finite. Minimal Polynomials Let E be a finite extension of F q. Then E is a vector space over F q and so E= F q t for some positive integer t. By Theorem , each element α of E is a root of the polynomial x qt x. Thus there is a monic polynomial M α in F q [x] of smallest degree which has α as a root, this polynomial is called the minimal polynomial of α over F q. In the following theorem we collect some elementary facts about minimal polynomials. 16
23 Definition [19] A minimal polynomial of an element α F q m with respect to F q is a nonzero monic polynomial f(x) of the least degree such that f(α) = 0. Theorem [20] Let F < E be fields, and let α E have minimal polynomial m(x) over F. 1) The polynomial m(x) is the unique monic irreducible polynomial over F for which m(α) = 0. 2) The polynomial m(x) is the unique monic polynomial of smallest degree over F for which m(α) = 0. 3) The polynomial m(x) is the unique monic polynomial over F with property that, for all f(x) F[x], we have f(α) = 0 if and only if m(x) f(x). Definition [19] Let n be coprime to q. The cyclotomic coset of q (or q-cyclotomic coset) modulo n containing i is defined by C i = {(i. q j (mod n) Z n : j = 0, 1,... }. A subset {i 1,..., i t } of Z n is called a complete set representatives of cyclotomic cosets of q modulo n if C i1,..., C it are distinct and t j C i j = Z n. Example [19] Consider the cyclotomic cosets of 2 modulo 15: C 0 = {0}, C 1 = {1, 2, 4, 8}, C 3 = {3, 6, 9, 12}, C 5 = {5, 10}, C 7 = {7, 11, 13, 14}. Thus, C 1 = C 2 = C 4 = C 8, and so on. The set {0, 1, 3, 5, 7} is complete set of representatives of cyclotomic cosets of 2 modulo 15. Example [24] The polynomial f(x) = 1 + x + x 3 is irreducible over F 2 ; if it were reducible, it would have a factor of degree 1 and hence a root in F 2, which it does not. So F 8 = F 2 / < f(x) >, The elements of F 8 are given by: Cosets V ectors P olynomials in α P ower of α 0+ < f(x) > < f(x) > = α 0 x+ < f(x) > 010 α α x + 1+ < f(x) > 011 α + 1 α 3 x 2 + < f(x) > 100 α 2 α x 2 + < f(x) > 101 α α 6 x 2 + x+ < f(x) > 110 α 2 + α α 4 x 2 + x + 1+ < f(x) > 111 α 2 + α + 1 α 5 17
24 The column power of α is obtained by using f(α) = α 3 + α + 1 = 0, which implies that α 3 = α + 1. So α 4 = αα 3 = α(α + 1) = α 2 + α, α 5 = αα 4 = α(α 2 + α) = α 3 + α 2 = α 2 + α + 1, etc. Example [24] The field F 8 was constructed in the Example above. In the table below we give the minimal polynomial over F 2 of each element of F 8 and the associated 2-cyclotomic coset modulo 7. Roots M inimal polynomial 2 cyclotomic coset 0 x x {0} α, α 2, α 4 x 3 + x + 1 {1, 2, 4} α 3, α 5, α 6 x 3 + x {3, 5, 6} 1.2 Basic definitions of coding theory Coding theory deals with the problem of detecting and correcting transmission errors caused by noise on the channel. In many cases, the information to be sent is transmitted by a sequence of zeros and ones. We call a 0 or a 1 a digit. A word is a sequence of digits. The length of a word is the number of digits in the word. Thus is a word of length seven. A word is transmitted by sending its digits, one after the other, across a binary channel. The term binary refers to the fact that only two digits 0 and 1 are used. Each digit is transmitted mechanically, electrically, magnetically, or otherwise by one of two types of easily differentiated pluses. A binary code is a set C of words over Z 2. The code consisting of all words of length two is C = {00, 10, 01, 11}. A block code is a code having all its words of the same length; this number is called the length of a code. The following diagram provides a rough idea of a general information transmission system. Information Source Transmitter (Encoder) Communication Channel Receiver (Decoder) Information Sink noise 18
25 The most important part of diagram, as far as we are concerned, is the noise, for without it there would be no need for the theory. In practice, the control we have over this noise is the choice of a good channel to use for transmission and the use of various noise filters to combat certain types of interference which may be encountered. These are engineering problems. Once we have settled on the best mechanical system for solving these problems, we can focus our attention on the construction of the encoder and decoder. Our desire is to construct these in such a way as to effect: 1) Fast encoding of information 2) Easy transmission of encoded messages 3) Fast decoding of received messages 4) Correction of errors introduced in the channel, and 5) Maximum transfer of information per unit time. Here we will define the terminology that we will use throughout the thesis. Strings and codes Definition [7] Let A = {a 1, a 2...a v } be a finite set of v elements. A v-ary code C of length n is a non empty subset of n-tuples with entries in A i.e., C (A) n The elements of the code C are called codewords, and C is called a v-ary block code. The size v of the code alphabet is called the radix of the code. The code C depends on v, a code whose alphabet is Z 2 = {0, 1} is called a binary code or a Z 2 -code, a code whose alphabet is Z 3 = {0, 1, 2} is called a ternary code or a Z 3 -code and a code whose alphabet consists of four elements such as Z 4 = {0, 1, 2, 3} is called quaternary code or a Z 4 -code. We denote the number of the codewords in a code C by C. If C A contains M codewords, then we say that C has length n and size M, and we denote it by (n, M)-code. 19
26 Definition [7] The (Hamming distance) d(x, y) between two vectors x, y F n q is defined to be the number of coordinates in which x and y differ. The (Hamming weight) wt(x) of a vector x F n q is the number of nonzero coordinates in x. Definition [24] For a code C containing at least two words, the minimum distance of a code C, denoted by d(c), is d(c) = min{d(x, y) : x, y C, x y}. Theorem [24] If x, y F n q, then d(x, y) = wt(x y). If C is a linear code, the minimum distance d is the same as the minimum weight of the nonzero codewords of C Theorem [24] The distance function d(x, y) satisfies the following four properties: (i) (non-negativity) d(x, y) 0 for all x, y F n q. (ii) d(x, y) = 0 if and only if x = y. (iii) (symmetry) d(x, y) = d(y, x) for all x, y F n q. (iv) (triangle inequality) d(x, z) d(x, y) + d(y, z) for all x, y, z F n q. Example [19] Let C = {00000, 00111, 11111} be binary code. Then d(c) = 2 since d(00000, 00111) = 3, d(00000, 11111) = 5, d(00111, 11111) = 2. Hence, C is a binary (5, 3, 2)-code. Theorem [24] A code with distance d is an exactly (d 1) error detecting code. Definition (Linear codes over fields ) Let the alphabet F q be the Galois Field of q elements. A q-array linear code of length n and dimension k is a linear subspace C F n q of the vector space of dimension n. Namely; for every c 1, c 2 C and a 1, a 2 F we have a 1 c 1 + a 2 c 2 C. If C has minimum distance d, then we record that the parameters of C over F q as [n, k, d] q. Definition [24] A generator matrix for an [n, k] code C is any k n matrix G whose rows form a basis for C. Note that a generator matrix for C must have k rows and n columns, and it must have rank k. 20
27 Definition [24] A generator matrix of the form [I k A] where I k is the k k identity matrix is said to be in the standard form. Theorem [24] If G = [I k A] is a generator matrix for the [n, k] code C is standard form, then H = [ A τ I n k ] is a parity check matrix for C. A matrix H is called a parity-check matrix for a linear code C if the rows of H form a basis for the dual code C. If C has length n and dimension k, then the sum of the dimensions of C and C is n, any parity-check matrix for C must have n rows, n k columns and rank n k. Definition [24] Let A i, also denoted A i (C), be the number of codewords of weight i in C. The list A i for 0 i n is called the weight distribution or weight spectrum of C. Example [24] Let C be binary code with generator matrix G = The weight distribution of C is A 0 = A 6 = 1 and A 2 = A 4 = 3. Notice that only the nonzero A i are usually listed. Definition [24] The single variable weight enumerator of C is W c (x) = n A i (C)x i. i=0 By replacing x by x/y and then multiplying by y n, W c (x) can be converted to the two variable weight enumerator W c (x, y) = n A i (C)x i y n i. i=0 Where the list A i for 0 i n is called the weight distribution. By Example 1.2.2, the two variable weight enumerator of C is W c (x, y) = n A i (C)x i y n i = y 6 + 3x 2 y 4 + 3x 4 y 2 + x 6. i=0 21
28 Codes over rings The study of linear codes over finite rings has received much attention lately and many recent developments of coding theory are defined on finite rings in particular over rings of four alphabets. For the purpose of this thesis we will consider alphabets as rings under addition and multiplication. Quaternary codes Let Z 4 denote the integers modulo 4. Z 4 is a ring which has 2 as a zero divisor. Definition A set C of n-tuples over Z 4 is called a code over Z 4 or a Z 4 - code. If C is a Z 4 module we say that C is a linear code over Z 4 or a quaternary code. Example [7] The quaternary code; is a linear code. C = {000, 010, 020, 030, 202, 212, 222, 232} Example [7] For the code, C = {000, 011, 203} is not linear code since 011 and 203 are in C but = 210 is not in C. Definition [24] The inner product of vectors x = x 1... x n, y = y 1... y n in Fq n is The C is defined by x.y = n i=1 x iy i. C = {x F n q : x.c = 0, c C}. Definition [24] A code C is called self-orthogonal provided C C. Definition [24] A code C is called self-dual if C = C and the length n of a self-dual code is even and the dimension is n/ Cyclic Codes Over Finite Fields One of the most important classes of linear codes are the class of cyclic code. These codes have great practical importance and they are also of considerable interest from an algebraic point of view since they are easy to encode. They also include the important family Bose-Chadhuri-Hocquengham (BCH) codes 22
29 which are great practical importance for error correction, particulary the number of errors is expected to be small compared with the length of the code. Also cyclic codes are considered important since they are the building blocks for many other codes. We assume throughout our discussion of cyclic codes that n and q are relatively prime. In particular, if q = 2 then n must be odd. When examining cyclic codes over F q, we will most often represent the codewords in polynomial form. There is bijective correspondence between the vectors c = c 0 c 1... c n 1 in F n q and the polynomials c(x) = c 0 +c 1 x+... c n 1 x n 1 in F q [x] of degree at most n 1. Notice that if c(x) = c 0 + c 1 x +... c n 1 x n 1, then xc(x) = c n 1 x n + c 0 x + c 1 x c n 2 x n 1, which would represent the codeword c cyclically shifted one to the right if x n were set equal to 1. More formally, the fact that a cyclic code C is invariant under a cyclic shift implies that if c(x) is in C, then so is xc(x) provided we multiply modulo x n 1. Polynomials and Words The polynomial f(x) = a 0 + a 1 x + a 2 x a n 1 x n 1 n 1 over field K may regarded as the word v = a 0 a 1 a 2... a n 1 of length n in K n. For example if n = 7, of degree at most polynomial word 1 + x + x 2 + x x 4 + x 5 + x x + x Thus a code of length n can be represented as a set of polynomials over K of degree at most n 1. The word a 0 a 1 a 2 a 3 of length 4 is represented by the polynomial a 0 + a 1 x + a 2 x 2 + a 3 x 3 of degree 3, for instance. Definition [7] Let υ be a word of length n, the cyclic shift π(υ) is the word of length n π(υ 0, υ 1,..., υ n 1 ) = (υ n 1, υ 0,..., υ n 2 ). 23
30 Definition [7] A code C is said to be cyclic if π(υ) C, whenever υ C. Example C 1 = {102, 210, 021, 201, 120, 012, 222, 111, 000} is a linear cyclic code over Z 3, but C 2 = {000, 221, 212, 200, 121, 112, 100, 021, 012} is not cyclic since π(112) = 211 which is not in C 2 Theorem [24] If C 1 and C 2 are cyclic codes of length n over F q, then (i) C 1 + C 2 = {c 1 + c 2 : c 1 C 1, c 2 C 2 } is cyclic. (ii) C 1 C2 is cyclic. We remember that since F q [x] is principle ideal domain also the ring R n = F q [x]/< x n 1 > is a principle ideal hence the cyclic codes are principle ideals of R n when writing a code word of a cyclic code as c(x) we mean the coset c(x)+ < x n 1 > in R n. Corollary [24] The number of cyclic codes of cyclic codes in R n equal 2 m, where m is the number of q-cyclotomic cosets modulo n. Moreover, the dimensions of cyclic codes in R n are all possible sums of the sizes of the q- cyclotomic cosets modulo n. Generating polynomial of a cyclic code Theorem [15] A linear code C in F q is cyclic C is an ideal in R n = F q [x]/(x n 1). Proof. If C is an ideal in F q [x]/(x n 1) and c(x) = c 0 + c 1 x c n 1 x n 1 is any codeword, then xc(x) is also a codeword, i.e (c n 1, c 0, c 1,... + c n 2 ) C. Conversely, if C is cyclic, then c(x) C we have xc(x) C. Therefore x i c(x) C, and since C is linear, then a(x)c(x) C for each polynomial a(x). Hence C is an ideal. Theorem [24] Let C be an ideal in R n, then (i) There is a unique monic polynomial g(x) of minimum degree in C =< g(x) >, and it is called the generating polynomial for C. (ii) The generating polynomial g(x) divides x n 1. (iii) If deg(g(x)) = r, then C has dimension n r and C =< g(x) >= {s(x)g(x) : deg s(x) < n r}. 24
31 (iv) If g(x) = g 0 + g 1 x g r x r, then g 0 0 and C has the following generator matrix: g 0 g 1 g 2... g r g 0 g 1 g 2... g r G = 0 0 g 0 g 1 g 2... g r g 0 g 1 g 2. g r Proof. (i) Suppose that C contains two distinct monic polynomial g 1 and g 2 of minimum degree r. Then their difference g 1 g 2 would be a nonzero polynomial in C of degree less than r, which is not possible. Hence, there is a unique monic polynomial g(x) of degree r in C. Since g(x) C and C is an ideal, we have < g(x) > C. On the other hand, Suppose that p(x) C, and let p(x) = q(x)g(x) + r(x) where r(x) 0 and deg(r(x)) < r. Then r(x) = p(x) q(x)g(x) C has degree less than r, which possible only if r(x) = 0. Hence p(x) = q(x)g(x) < g(x) >, and so C < g(x) >. Thus C =< g(x) >. (ii) Dividing x n 1 by g(x) gives x n 1 = q(x)g(x) + r(x), where deg(r(x)) < r. Since in R n, we see that r(x) C, and so r(x) = 0, which shows that g(x) (x n 1). (iii) The ideal generated by g(x) is < g(x) >= {f(x)g(x) : f(x) R n } with the usual reduction mod (x n 1). Now f(x) divides x n 1, and so x n 1 = h(x)g(x) for some h(x) of degree n r. Divide f(x) by h(x), we get f(x) = q(x)h(x) + s(x), where deg(s(x)) < n r, then f(x)g(x) = q(x)g(x)h(x) + s(x)g(x) = q(x)(x n 1) + s(x)g(x). So f(x)g(x) = s(x)g(x) C. Now let c(x) be in C, then c(x) = s(x)g(x) = (a 0 + a 1 x + a 2 x a n r 1 x n r 1 )g(x) = (a 0 g(x) + a 1 xg(x) a n r 1 x n r 1 g(x). So c(x) < {g(x), xg(x),..., x n r 1 g(x)} >, which shows that the set {g(x), xg(x),..., x n r 1 g(x)} spans C. Also {g(x), xg(x),..., x n r 1 g(x)} is linearly independent, since if 25
32 a 0 g(x) + a 1 xg(x) a n r 1 x n r 1 g(x) = 0, then (a 0 + a 1 x + a 2 x a n r 1 x n r 1 )g(x) = 0 which implies that (a 0 + a 1 x + a 2 x a n r 1 x n r 1 ) = 0, and since 1, x, x 2,..., x n r 1 are linearly independent, then a 0 = a 1 =... = a n r 1 = 0 and hence {g(x), xg(x),..., x n r 1 g(x)} forms a basis for C. Hence dim(c) = n r. (iv) If g 0 = 0 then g(x) = xg 1 (x), where deg(g 1 (x)) < r and g 1 (x) = 1.g 1 (x) = x n 1 g(x), so g 1 (x) C which contradict the fact that no nonzero polynomial in C has degree less than r. Thus g 0 0. Finally, G is a generator matrix of C since {g(x), xg(x),..., x n r 1 g(x)} is a basis for C. Theorem [24] A monic polynomial p(x) in R n is the generator polynomial for an ideal p(x) (x n 1). Proof. (= ) Was proved by the previous theorem. ( =) Assume that p(x) (x n 1), let g(x) be the generator polynomial for an ideal C that contains p(x), then p(x) = a(x)g(x), where deg(a(x)) < n r. By the previous theorem g(x) (x n 1), so x n 1 = g(x)h(x), where h(x) is the check polynomial for C. By assumption x n 1 = p(x)h(x), for some polynomial h(x). So x n 1 = p(x)h(x) = a(x)g(x)h(x) = a(x)(x n 1). So a(x) = 1 and therefore, p(x) = g(x). The Parity Check Matrix Since the generating polynomial g(x) of a cyclic [n, n r]-code in R n divides x n 1, says x n 1 = g(x)h(x), then h(x) is a polynomial of degree n r, called the parity check polynomial of C. Theorem [20] Let h(x) be the check polynomial for a cyclic code C in R n, then (i) The code C can be described by C = {p(x) R n : p(x)h(x) = 0}. (ii) If h(x) = h 0 + h 1 x + h 2 x h n r x n r, then the parity check matrix for C is given by 26
33 h n r h h n r h H =... 0 h n r h h n r h 0 Example Let C be a cyclic code of length n = 9. Since x 9 1 factors over F 2 x 9 1 = (x 3 1)(x 6 + x 3 + 1) = (x 1)(x 2 + x + 1)(x 6 + x 3 + 1). Hence, there are 2 3 = 8 cyclic codes in R 9 = F 2 / < x 9 1 >. Take C =< x 6 + x > with generating polynomial g(x) = x 6 + x Then C has dimension 9 6 = 3 and generating matrix G = Also C has check polynomial h(x) = x9 1 g(x) = (x 1)(x2 + x + 1) = x 3 1. Then C has the parity check matrix H = Idempotents For Linear Codes We note that all cyclic codes can be obtained from factorization of x n 1 into monic irreducible factors over F q. However, factoring x n 1 is not so easy in general. In fact there are other generators that can be found without factoring x n 1, and they give another approach to describe cyclic codes. These are called idempotent generators. Definition [7] A polynomial e(x) R n is called an idempotent in R n if e 2 (x) e(x). Example [24] In R 7 = F 2 [x]/ < x 7 1 >, the polynomial x 3 + x 5 + x 6 is an idempotent since (x 3 + x 5 + x 6 ) 2 = x 3 + x 5 + x 6. 27
34 Definition [23] A ring R is called a Boolean ring if x 2 = x for all x R. Theorem [23]If every element in a ring R is idempotent, then R is commutative ring. Theorem [20] Let C be a cyclic code in R n with generator polynomial g(x) and check polynomial h(x). Then g(x) and h(x) are relatively prime and so there exist polynomial a(x) and b(x) for which a(x)g(x) + b(x)h(x) = 1 (*) The polynomial e(x) = a(x)g(x) mod (x n 1) has the following properties: 1) e(x) is the unique identity in C, that is p(x)e(x) p(x) p(x) C 2) e(x) is the unique polynomial in C that is both idempotent and generates C, that is C =< e(x) >. Proof. If e 1 (x) and e 2 (x) are both identities in R, then e 1 (x) e 1 (x)e 2 (x) e 2 (x) and so e 1 (x) = e 2 (x). Thus if an identity exists, then it is unique. Since g(x)h(x) = x n 1 has no multiple roots in any extension field, g(x) and h(x) are relatively prime, and so (*) holds. If p(x) C, then p(x)h(x) 0, see Theorem and (*) gives a(x)g(x)p(x) p(x) which says that e(x) a(x)g(x) mod(x n 1) is indeed the identity in C and also that e(x) generates C, since any polynomial in C is a multiple of e(x). Multiplying (*) by a(x)g(x) gives [a(x)g(x)] 2 + a(x)b(x)g(x)h(x) = a(x)g(x) and [a(x)g(x)] 2 a(x)g(x) thus, e(x) is an idempotent. To complete the proof, need only shows that an idempotent f(x) that also generates C must equal e(x). Since f(x) generates C, there exists q(x) R n for which e(x) q(x)f(x). Hence f(x) e(x)f(x) q(x)f 2 (x) q(x)f(x) e(x) which implies that f(x) = e(x). Thus completes the proof. 28
35 Theorem [20] The generator polynomial of the code < e(x) > is g(x) = gcd(e(x), x n 1) Proof. By the previous theorem, since x n 1 = g(x)h(x) and e(x) a(x)g(x), we have gcd(e(x), x n 1) = gcd(a(x)g(x), h(x)g(x)) = g(x). Example [24] The following table gives all the cyclic codes C i of length 7 over F 2 together with their generator polynomials g i (x) and their generating idempotents e i (x). i dim g i (x) e i (x) x x + x x x + x x x 2 + x 3 + x x 3 + x 5 + x x + x 2 + x x + x 2 + x x + x 3 x + x 2 + x x 2 + x 3 x 3 + x 5 + x x x + x x The two codes of dimension 4 are [7, 4, 3] Hamming codes. Example [24] The following table gives all the cyclic codes C i of length 11 over F 3 together with their generator polynomials g i (x) and their generating idempotents e i (x). i dim g i (x) e i (x) x x + x x 10 1 x x 2... x x x 2 x 3 + x 4 + x x + x 3 + x 4 + x 5 + x x 2 x 3 x 4 x 5 + x x 2 + x 6 + x 7 + x 8 + x x 2 x 3 + x 4 + x 5 x 2 x 6 x 7 x 8 x x + x 2 x 3 + x 5 x x 3 x 4 x 5 x x 1 + x + x x The two codes of dimension 6 are [11, 6, 5] Hamming codes. Theorem [20] Let C 1 and C 2 be cyclic codes with corresponding generators g 1 (x) and g 2 (x), and corresponding idempotent generators e 1, e 2, then (i) C 1 C2 has idempotent e 1 e 2 (ii) C 1 + C 2 has idempotent e 1 + e 2 e 1 e 2. Proof. (i) e 1 (x)e 2 (x) C 1 C2 and (e 1 (x)e 2 (x)) 2 = (e 1 (x)) 2 (e 2 (x)) 2 = e 1 (x)e 2 (x). 29
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